Training Keras Towards Or Against Analog Value?
What is the technical name for this kind of training?

The name for the problem is Sequential Decision Making or Optimal Control.
There are a few different approaches you can take when solving this kind of problem. However, I think that the way that you are describing your project, Reinforcement Learning (RL) would match your approach the best.

I cannot use a traditional Xy method but need another method that goes like this:
network.train_against_value(X, y, determinator)

Although this method signature could probably be made to work by re-structuring one or other RL frameworks, it is more usual when using RL with neural networks to treat the experience-gathering and scoring systems - the core ideas behind RL - as a data generator for either a supervised learning problem or in some cases directly producing the output layer gradients.
One approach you could use is called Deep Q Networks (DQN) which effectively generates mini-batches for supervised learning of a neural network, based on gathering experiences.
In brief, DQN trains a neural network to predict what you are calling the &quot;determinator&quot;, but that RL would call the &quot;value&quot; of each action. So you may move the action choice (y) to the input of the neural network, or alternatively predict multiple values (one output for each possble action) - the RL theory is the same for each approach here, it is an implementation detail. The agent (cat or mouse) would then pick the action with the highest predicted value by default (with the cat's values being the negative of the mouse's), trying other actions at random whilst training so as to learn all values. Whenever the agent gained new experience, it would add that to the training data so it could improve its predictions.
RL is a complex subject in its own right, and to begin understanding it properly you would do well to start by tackling even simpler problems that don't require neural networks or running two agents against each other. There is a good introductory book on the subject with the option to read a free PDF version: Reinforcement Learning: An Introduction (Sutton &amp; Barto)